Let G he a finitely generated group, and A a Z [G]-module of flat dimension n such that the homological invariant Sigma(n)(G, A) is not empty. We show that A has projective dimension n as a Z[G]-module. In particular, if G is a group of homological dimension hd(G) = n such that the homological invariant Sigma(n)(G, Z) is not empty, then G has cohomological dimension cd(G) = n. We show that if G is a finitely generated soluble group, the converse is true subject to taking a subgroup of finite index, i.e., the equality cd(G) = hd(G) implies that there is a subgroup H of finite index in G such that Sigma(infinity)(H, Z) not equal 0.351253259